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Regions surrounded by cylinders of real algebraic manifolds and natural decompositions

Naoki Kitazawa

TL;DR

This work advances the understanding of regions in $\mathbb{R}^k$ bounded by cylinders of real algebraic hypersurfaces by formalizing RA-regions and analyzing their structure through singularity theory. The main contribution is a theorem showing that, under pre-real-algebraic-region (PRAR) conditions and a $b$-intersection property for a coordinate cover ${A_i}$, the intersection of projected domains $\bigcap_{i=1}^a \pi_{n,A_i}^{-1}(D_{A_i})$ yields an RA-region. It also provides corollaries for small coordinate subsets, extends the framework to $N_k$-singular/normal point scenarios, and connects to higher-dimensional generalizations, while situating the work within explicit real algebraic map construction and Reeb-graph reconstruction. The results offer a rigorous, constructive pathway to decompose and realize real algebraic objects with controlled topology, linking real algebraic geometry, singularity theory, and topological encoding via Reeb graphs.

Abstract

The author has been interested in regions surrounded by cylinders of real algebraic hypersurfaces and their shapes and polynomials associated to them. Here, we formulate and investigate natural decompositions into such cylinders of real algebraic hypersurfaces. Especially, intersections of these cylinders of real algebraic hypersurfaces, which give important information on regions, are investigated via singularity theory. This is a kind of natural problems on real geometry. This also comes from construction of explicit real algebraic maps onto explicit regions in real affine spaces on real algebraic manifolds. More generally, we are interested in difficulty in explicit construction of real algebraic objects, where existence and approximation has been well-known, since pioneering studies by Nash and Tognoli, in the latter half of 20th century. This also comes from interest in singularity theory of differentiable, smooth or real algebraic functions and maps, especially, explicit construction.

Regions surrounded by cylinders of real algebraic manifolds and natural decompositions

TL;DR

This work advances the understanding of regions in bounded by cylinders of real algebraic hypersurfaces by formalizing RA-regions and analyzing their structure through singularity theory. The main contribution is a theorem showing that, under pre-real-algebraic-region (PRAR) conditions and a -intersection property for a coordinate cover , the intersection of projected domains yields an RA-region. It also provides corollaries for small coordinate subsets, extends the framework to -singular/normal point scenarios, and connects to higher-dimensional generalizations, while situating the work within explicit real algebraic map construction and Reeb-graph reconstruction. The results offer a rigorous, constructive pathway to decompose and realize real algebraic objects with controlled topology, linking real algebraic geometry, singularity theory, and topological encoding via Reeb graphs.

Abstract

The author has been interested in regions surrounded by cylinders of real algebraic hypersurfaces and their shapes and polynomials associated to them. Here, we formulate and investigate natural decompositions into such cylinders of real algebraic hypersurfaces. Especially, intersections of these cylinders of real algebraic hypersurfaces, which give important information on regions, are investigated via singularity theory. This is a kind of natural problems on real geometry. This also comes from construction of explicit real algebraic maps onto explicit regions in real affine spaces on real algebraic manifolds. More generally, we are interested in difficulty in explicit construction of real algebraic objects, where existence and approximation has been well-known, since pioneering studies by Nash and Tognoli, in the latter half of 20th century. This also comes from interest in singularity theory of differentiable, smooth or real algebraic functions and maps, especially, explicit construction.
Paper Structure (4 sections, 5 theorems)

This paper contains 4 sections, 5 theorems.

Table of Contents

  1. Introduction.
  2. On our main result.
  3. Additional notes.
  4. Conflict of interest and data availability.

Key Result

Theorem 1

Let $\{S_j\}_{j=1}^{l}$ be a family of real algebraic manifolds of dimension $n-1$ in ${\mathbb{R}}^n$ and $\{A_i\}_{i=1}^a$ a family of $a \in \mathbb{N}$ distinct subsets of ${\mathbb{N}}_n$ such that $A_{i_1} \subset A_{i_2}$ does not hold for any pair of distinct integers $1 \leq i_1, i_2 \leq n Under the present situation, $({\bigcap}_{i=1}^a {{\pi}_{n,A_i}}^{-1}(D_{A_i}),\{S_j\}_{j=1}^l)$ is

Theorems & Definitions (7)

  • Theorem 1
  • proof : A proof of Theorem \ref{['thm:1']}
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • proof
  • Theorem 4: kitazawa4